Be careful that you are using the right type of pocket: either for bag or vault space. If you have purchased the wrong type, it can be traded after a time limit has passed (see in-game info for the time limit).

Make sure that your bag or vault space is not at the maximum size. The maximum size for a bag or vault is listed in the information about the pocket (in the Showcase information window).

Inventory - which explains the general topic of personal inventory, but includes info on other ways to store items and how to split up stacks of materials and other items (eg. how to take 1 item out of a stack of 100)

Vault - will tell you how you can store items and access them from anywhere, much like a bank in other games.